Counting spinal phylogenetic networks

TL;DR

This paper introduces a combinatorial approach using expanding covers to count and analyze specific classes of phylogenetic networks, including spinal networks, which are related to well-known classes like orchard and tree-sibling networks.

Contribution

It presents a novel method for counting spinal phylogenetic networks using expanding covers, providing a foundation for enumerating more complex network classes.

Findings

Expanding covers effectively encode labellable networks.

A counting method for spinal networks is developed.

Potential for deriving formulas for various network classes.

Abstract

Phylogenetic networks are an important way to represent evolutionary histories that involve reticulations such as hybridization or horizontal gene transfer, yet fundamental questions such as how many networks there are that satisfy certain properties are very difficult. A new way to encode a large class of networks, using expanding covers, may provide a way to approach such problems. Expanding covers encode a large class of phylogenetic networks, called labellable networks. This class does not include all networks, but does include many familiar classes, including orchard, normal, tree-child and tree-sibling networks. As expanding covers are a combinatorial structure, it is possible that they can be used as a tool for counting such classes for a fixed number of leaves and reticulations, for which, in many cases, a closed formula has not yet been found. More recently, a new class of networks was introduced, called spinal networks, which are analogous to caterpillar trees for phylogenetic trees and can be fully described using covers. In the present article, we describe a method for counting networks that are both spinal and belong to some more familiar class, with the hope that these form a base case from which to attack the more general classes.